# On Diamond's $L^1$ criterion for asymptotic density of Beurling generalized integers

## TL;DR

This paper provides a simplified proof of the $L^1$ criterion ensuring positive asymptotic density for Beurling generalized integers, and explores weaker conditions for density existence and related estimates.

## Contribution

It offers a shorter proof of the $L^1$ criterion and introduces weaker hypotheses for the existence of asymptotic density in Beurling integers.

## Key findings

- Shorter proof of the $L^1$ criterion for positive density.
- Weaker hypotheses can ensure the existence of density.
- Discussion of conditions for the estimate involving Beurling's Möbius function.

## Abstract

We give a short proof of the $L^{1}$ criterion for Beurling generalized integers to have a positive asymptotic density. We actually prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)=\sum_{n_{k}\leq x} \mu(n_k)/n_k=o(1)$, with $\mu$ the Beurling analog of the Moebius function.